We begin the master thesis with a short overview of some famous equations of mathematical physics. We recall some of the basic methods for solving partial differential equations and present their traveling wave solutions. We continue by focusing on the Korteweg-deVries equation and the corresponding Lax pair. Using this important mathematical tool we derive the N-soliton solution of the Korteweg-deVries equation. We also present the Toda lattice, a system of ordinary differential equations, which we solve with the help of a Lax pair and a method, called the inverse scattering transform. The thesis is concluded by the detailed presentation of nonlinear Schrödinger equation and corresponding matrix Riemann-Hilbert problem.